When all coefficients of and are real (implying that is the transfer function of a real filter), it will always happen that the complex one-pole filters will occur in complex conjugate pairs. Let denote any one-pole section in the PFE of Eq. (6.7). Then if is complex and describes a real filter, we will also find somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:
Expressing the pole in polar form as , and the residue as , the last expression above can be rewritten as
The use of polar-form coefficients is discussed further in the section on two-pole filters (§B.1.3).
Expanding a transfer function into a sum of second-order terms with real coefficients gives us the filter coefficients for a parallel bank of real second-order filter sections. (Of course, each real pole can be implemented in its own real one-pole section in parallel with the other sections.) In view of the foregoing, we may conclude that every real filter with can be implemented as a parallel bank of biquads.^{7.6} However, the full generality of a biquad section (two poles and two zeros) is not needed because the PFE requires only one zero per second-order term.
To see why we must stipulate in Eq. (6.7), consider the sum of two first-order terms by direct calculation:
(7.9) |